sage: H = DirichletGroup(83)
pari: g = idealstar(,83,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 82 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{82}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{83}(2,\cdot)$ |
First 32 of 82 characters
Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.
Character | Orbit | Order | Primitive | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{83}(1,\cdot)\) | 83.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{83}(2,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{82}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{73}{82}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) |
\(\chi_{83}(3,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) |
\(\chi_{83}(4,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) |
\(\chi_{83}(5,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{73}{82}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) |
\(\chi_{83}(6,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{73}{82}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{55}{82}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) |
\(\chi_{83}(7,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) |
\(\chi_{83}(8,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{55}{82}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{9}{82}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) |
\(\chi_{83}(9,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) |
\(\chi_{83}(10,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) |
\(\chi_{83}(11,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) |
\(\chi_{83}(12,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) |
\(\chi_{83}(13,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{77}{82}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{29}{82}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{67}{82}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) |
\(\chi_{83}(14,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{9}{82}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{79}{82}\right)\) | \(e\left(\frac{1}{82}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) |
\(\chi_{83}(15,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{82}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{49}{82}\right)\) | \(e\left(\frac{11}{82}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{51}{82}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) |
\(\chi_{83}(16,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) |
\(\chi_{83}(17,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) |
\(\chi_{83}(18,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{7}{82}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{25}{82}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) |
\(\chi_{83}(19,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{47}{82}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{39}{82}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{24}{41}\right)\) | \(e\left(\frac{59}{82}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{2}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) |
\(\chi_{83}(20,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{29}{82}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{67}{82}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{5}{82}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) |
\(\chi_{83}(21,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) |
\(\chi_{83}(22,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{25}{82}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{19}{82}\right)\) | \(e\left(\frac{21}{82}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{75}{82}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) |
\(\chi_{83}(23,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) |
\(\chi_{83}(24,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{75}{82}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{57}{82}\right)\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) |
\(\chi_{83}(25,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) |
\(\chi_{83}(26,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) |
\(\chi_{83}(27,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) |
\(\chi_{83}(28,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) |
\(\chi_{83}(29,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{3}{41}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) |
\(\chi_{83}(30,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{33}{41}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{1}{41}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{11}{41}\right)\) |
\(\chi_{83}(31,\cdot)\) | 83.c | 41 | yes | \(1\) | \(1\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{40}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) |
\(\chi_{83}(32,\cdot)\) | 83.d | 82 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{82}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{15}{82}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{19}{41}\right)\) |